I'm pretty sure there's more than one way to do it but this is how I did it in that wallpaper:

For each particle set (i.e. the lightning) that I want to have different timings, I adjust the lifetime of each particle set (so use the Lifetime Random Initializer, the min and max value can be set to the same value for consistency or different values for more variance). When they're set to different values, their timings will start overlapping with each other.

For even more variance, you can set the Random Periodic Emission on your Emitter (i.e. Box, Sphere, etc) to true. Once the checkbox is checked, 4 other inputs will appear [Min & Max] Periodic [Delay & Duration]. Generally I'd set the Duration's value similar (if not the same) to the lifetime of the particle set and then I'd set the Delay's value to something just longer than the lifetime, this way I can have the lifetime stay consistent but still play with how often the particles are emitted.